Tchebycheffian splines are smooth piecewise functions whose pieces are drawn from (possibly different) extended Tchebycheff (ET-) spaces which are natural generalizations of algebraic polynomial spaces [1]}, [2]}, [3]}.
Any non-trivial element of an ET-space of dimension \(p+1\)  has at most \(p\)  zeros counting multiplicity. We will refer to \(p\)  as the degree, in analogy with the polynomial case.
Extended complete Tchebycheff (ECT-) spaces are an important subclass that can be generated through a set of positive weight functions [4]}, [3]} and are spanned by generalized power functions [6]}; the latter are the natural extension of the monomial basis functions for algebraic polynomials; see sec-ECT-spaces. Relevant examples are null-spaces of linear differential operators on suitable intervals [3]}; see secnullspc. On bounded and closed intervals the concepts of ET-space and ECT-space coincide [8]}, so from now on we will focus on ECT-spaces.
